Mathematical constants by continued fraction representation: Difference between revisions
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In [[mathematics]], particularly in [[dynamical systems]], a '''bifurcation diagram''' shows the possible long-term values (equilibria/fixed points or periodic orbits) of a system as a function of a [[Bifurcation theory|bifurcation parameter]] in the system. It is usual to represent stable solutions with a solid line and unstable solutions with a dotted line. | |||
==Bifurcations in 1D discrete dynamical systems== | |||
===Logistic map=== | |||
{{See also|Dynamical systems|List of chaotic maps}} | |||
[[Image:LogisticMap BifurcationDiagram.png|300px|thumb|right|Bifurcation diagram of the [[logistic map]]]] | |||
[[Image:diagram bifurkacji anim small.gif|300px|thumb|right|Animation showing the formation of bifurcation diagram]] | |||
[[Image:Circle map bifurcation.jpeg|thumb|right|Bifurcation diagram of the [[circle map]]. Black regions correspond to [[Arnold tongues]].]] | |||
An example is the bifurcation diagram of the [[logistic map]]: | |||
:<math> x_{n+1}=rx_n(1-x_n). \,</math> | |||
The bifurcation parameter ''r'' is shown on the horizontal axis of the plot and the vertical axis shows the possible long-term population values of the logistic function. | |||
The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a [[period-doubling bifurcation]]. | |||
The ratio of the lengths of successive intervals between values of ''r'' for which bifurcation occurs [[convergent series|converges]] to the [[first Feigenbaum constant]]. | |||
===Real quadratic map=== | |||
{{See also|Complex quadratic polynomial}} | |||
The map is <math>x_{n+1}=x_n^2-c</math>. | |||
==Symmetry breaking in bifurcation sets== | |||
[[Image:Asymbif.gif|300px|thumb|right|Symmetry breaking in [[pitchfork bifurcation]] as the parameter epsilon is varied. epsilon = 0 is the case of symmetric pitchfork bifurcation.]] | |||
In a dynamical system such as | |||
:<math> \ddot {x} + f(x;\mu) + \epsilon g(x) = 0</math>, | |||
which is [[structurally stable]] when <math> \mu \neq 0 </math>, if a bifurcation diagram is plotted, treating <math> \mu </math> as the bifurcation parameter, but for different values of <math> \epsilon </math>, the case <math> \epsilon = 0</math> is the symmetric pitchfork bifurcation. When <math> \epsilon \neq 0 </math>, we say we have a pitchfork with ''broken symmetry.'' This is illustrated in the animation on the right. | |||
== See also == | |||
* [[Bifurcation theory]] | |||
* [[Feigenbaum constants]] | |||
* [[Phase portrait]] | |||
* [[Complex_quadratic_polynomial#Critical_curves|Skeleton of bifurcation diagram]] | |||
==References== | |||
*Paul Glendinning, "Stability, Instability and Chaos", Cambridge University Press, 1994. | |||
*Steven Strogatz, "Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering", Perseus Books, 2000. | |||
==External links== | |||
* [http://yuval.bar-or.org/index.php?item=4 Logistic Map Simulation]. A Java applet simulating the Logistic Map by Yuval Baror. | |||
* [http://www.egwald.com/nonlineardynamics/logisticsmapchaos.php The Logistic Map and Chaos] | |||
[[Category:Chaos theory]] | |||
[[Category:Bifurcation theory]] | |||
[[de:Bifurkationsdiagramm]] | |||
Revision as of 23:33, 26 January 2014
Template:No footnotes In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values (equilibria/fixed points or periodic orbits) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable solutions with a solid line and unstable solutions with a dotted line.
Bifurcations in 1D discrete dynamical systems
Logistic map
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An example is the bifurcation diagram of the logistic map:
The bifurcation parameter r is shown on the horizontal axis of the plot and the vertical axis shows the possible long-term population values of the logistic function.
The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a period-doubling bifurcation. The ratio of the lengths of successive intervals between values of r for which bifurcation occurs converges to the first Feigenbaum constant.
Real quadratic map
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The map is .
Symmetry breaking in bifurcation sets
In a dynamical system such as
which is structurally stable when , if a bifurcation diagram is plotted, treating as the bifurcation parameter, but for different values of , the case is the symmetric pitchfork bifurcation. When , we say we have a pitchfork with broken symmetry. This is illustrated in the animation on the right.
See also
References
- Paul Glendinning, "Stability, Instability and Chaos", Cambridge University Press, 1994.
- Steven Strogatz, "Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering", Perseus Books, 2000.
External links
- Logistic Map Simulation. A Java applet simulating the Logistic Map by Yuval Baror.
- The Logistic Map and Chaos